/*
  This code can be loaded, or copied and pasted, into Magma.
  It will load the data associated to the HMF, including
  the field, level, and Hecke and Atkin-Lehner eigenvalue data.
  At the *bottom* of the file, there is code to recreate the
  Hilbert modular form in Magma, by creating the HMF space
  and cutting out the corresponding Hecke irreducible subspace.
  From there, you can ask for more eigenvalues or modify as desired.
  It is commented out, as this computation may be lengthy.
*/

P<x> := PolynomialRing(Rationals());
g := P![-2, -8, 0, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {17, 17, -2*w^2 + 15}>;

primesArray := [
[2, 2, -w],
[3, 3, w^2 - 7],
[5, 5, w + 1],
[5, 5, -w - 3],
[9, 3, w^2 - 2*w - 1],
[17, 17, -2*w^2 + 15],
[17, 17, 3*w + 1],
[17, 17, -w^2 + w + 5],
[19, 19, -2*w^2 + w + 15],
[29, 29, w^2 - w - 1],
[41, 41, w^2 - 5],
[43, 43, w^2 + w - 3],
[47, 47, 2*w - 1],
[53, 53, -2*w - 3],
[59, 59, w^2 - w - 11],
[71, 71, w^2 - 3],
[73, 73, 6*w^2 - 2*w - 47],
[83, 83, w^2 - 4*w + 1],
[83, 83, 3*w^2 - 25],
[83, 83, w - 5],
[97, 97, -w^2 + w - 1],
[97, 97, 7*w^2 - 2*w - 57],
[103, 103, -2*w + 7],
[107, 107, -2*w^2 + 4*w + 3],
[107, 107, 3*w^2 - 2*w - 21],
[107, 107, w^2 + w - 9],
[113, 113, 5*w^2 - 12*w - 3],
[131, 131, -2*w^2 + 2*w + 13],
[139, 139, 2*w^2 - 19],
[149, 149, 3*w^2 - w - 21],
[163, 163, w^2 + 2*w - 5],
[167, 167, 3*w^2 - 7*w - 1],
[179, 179, 2*w^2 + w - 19],
[181, 181, -3*w - 5],
[191, 191, 4*w^2 - 2*w - 33],
[191, 191, w^2 - 2*w - 9],
[191, 191, 3*w^2 - 6*w - 1],
[193, 193, 2*w^2 - 1],
[197, 197, -2*w^2 + 5*w + 5],
[199, 199, w^2 - 3*w + 3],
[211, 211, -2*w^2 + 7*w + 3],
[223, 223, -3*w^2 + 6*w + 7],
[223, 223, w^2 - 3*w - 5],
[223, 223, 4*w^2 - 33],
[227, 227, -5*w^2 + 14*w + 5],
[229, 229, -4*w - 3],
[229, 229, 7*w^2 - w - 55],
[229, 229, -2*w^2 + 3*w + 3],
[239, 239, -2*w^2 - w - 1],
[251, 251, 2*w^2 - 5*w + 1],
[263, 263, 2*w^2 - 2*w - 17],
[263, 263, 3*w^2 - w - 27],
[263, 263, -2*w^2 + 9*w + 3],
[269, 269, w^2 - 5*w - 3],
[269, 269, 13*w^2 - 3*w - 105],
[269, 269, -3*w^2 + 7*w + 7],
[271, 271, 3*w^2 - 8*w - 5],
[281, 281, w^2 - w - 13],
[283, 283, -4*w^2 + 8*w + 3],
[293, 293, 2*w^2 + w - 17],
[307, 307, 5*w^2 - 2*w - 43],
[311, 311, -w^2 - w - 3],
[313, 313, w^2 - 5*w + 3],
[331, 331, 7*w^2 - 3*w - 53],
[343, 7, -7],
[349, 349, w^2 - 6*w - 3],
[353, 353, -3*w^2 + 3*w + 19],
[359, 359, 3*w^2 + 2*w - 15],
[361, 19, -4*w^2 + 10*w + 7],
[367, 367, 2*w^2 - 2*w - 7],
[367, 367, -3*w^2 + 19],
[367, 367, -w^2 - 3],
[373, 373, 9*w^2 - 2*w - 69],
[373, 373, w^2 + 2*w - 11],
[373, 373, 8*w^2 - 2*w - 61],
[383, 383, -4*w^2 + 9*w + 5],
[383, 383, 2*w^2 - w - 9],
[383, 383, 7*w^2 - w - 59],
[397, 397, 11*w^2 - 3*w - 85],
[401, 401, 2*w^2 - 3*w - 21],
[409, 409, 2*w^2 - 9],
[419, 419, 2*w^2 - 6*w + 3],
[419, 419, 2*w^2 + 2*w - 3],
[419, 419, -4*w^2 + 27],
[421, 421, -15*w^2 + 4*w + 121],
[421, 421, 3*w^2 - 1],
[421, 421, 2*w^2 - 2*w - 5],
[439, 439, 2*w^2 - 4*w - 9],
[449, 449, w^2 + 4*w - 3],
[449, 449, 5*w - 1],
[449, 449, 6*w^2 - w - 51],
[457, 457, -2*w - 9],
[457, 457, -3*w^2 + 5*w + 9],
[457, 457, 2*w^2 + w - 7],
[461, 461, 5*w^2 - 39],
[461, 461, w^2 - 3*w - 17],
[461, 461, 3*w^2 - 2*w - 29],
[463, 463, -7*w - 3],
[467, 467, 8*w^2 - 3*w - 63],
[487, 487, 8*w^2 - w - 63],
[499, 499, 6*w^2 - 2*w - 51],
[503, 503, w^2 - 6*w + 1],
[521, 521, 3*w^2 - 2*w - 17],
[521, 521, 2*w^2 - 3*w - 13],
[521, 521, w^2 - 6*w + 7],
[541, 541, -3*w^2 - w + 23],
[547, 547, -4*w^2 + 12*w + 5],
[557, 557, 6*w^2 - 14*w - 3],
[569, 569, 2*w^2 + 4*w - 7],
[571, 571, -2*w^2 - 2*w + 21],
[571, 571, 5*w^2 - 2*w - 35],
[571, 571, -3*w^2 + 9*w + 5],
[593, 593, -3*w - 11],
[599, 599, 3*w^2 - 4*w - 3],
[601, 601, -4*w^2 + 8*w + 9],
[607, 607, 2*w^2 - 8*w + 1],
[613, 613, -w^2 - 2*w - 5],
[617, 617, 3*w^2 - 29],
[619, 619, -5*w - 11],
[641, 641, w^2 + 2*w - 17],
[643, 643, 4*w - 5],
[647, 647, 5*w^2 - 3*w - 37],
[653, 653, 3*w^2 - 10*w + 5],
[653, 653, -5*w + 13],
[653, 653, 3*w^2 + 2*w - 19],
[659, 659, -w - 9],
[661, 661, w^2 - 4*w - 7],
[661, 661, -4*w - 13],
[661, 661, -3*w^2 + 6*w + 11],
[673, 673, -6*w^2 + 14*w + 9],
[677, 677, -3*w^2 + 5*w + 5],
[683, 683, 2*w^2 - 3*w - 19],
[709, 709, 4*w^2 - 3*w - 29],
[719, 719, 6*w - 1],
[727, 727, -w^2 - 7*w - 9],
[733, 733, -3*w^2 - 1],
[739, 739, -11*w^2 + 2*w + 91],
[743, 743, 3*w^2 + 3*w - 11],
[761, 761, 4*w^2 - 13*w + 1],
[769, 769, 12*w^2 - 2*w - 95],
[773, 773, -7*w^2 + 15*w + 5],
[787, 787, w^2 - 4*w + 7],
[811, 811, 8*w^2 - 2*w - 67],
[811, 811, 9*w + 1],
[811, 811, w^2 + 3*w - 9],
[821, 821, -5*w - 9],
[823, 823, 5*w - 3],
[839, 839, -w^2 + 7*w - 1],
[841, 29, 6*w^2 - w - 43],
[853, 853, 3*w^2 + 2*w - 13],
[853, 853, 3*w^2 - 3*w - 13],
[853, 853, -w^2 + w - 5],
[857, 857, 4*w^2 - 15*w - 3],
[859, 859, 3*w^2 + w - 15],
[877, 877, 3*w^2 - 3*w - 23],
[883, 883, -6*w - 5],
[883, 883, 9*w^2 - 2*w - 75],
[883, 883, w^2 - 6*w - 5],
[911, 911, 3*w^2 - 3*w - 25],
[919, 919, -5*w^2 + 11*w + 7],
[929, 929, 2*w^2 - 6*w - 7],
[937, 937, 4*w^2 - 3*w - 33],
[941, 941, 4*w^2 - 3*w - 39],
[947, 947, 6*w^2 - 21*w - 5],
[947, 947, 5*w^2 - 7*w - 3],
[947, 947, 6*w^2 - 3*w - 41],
[967, 967, w^2 - 4*w - 13],
[991, 991, 4*w^2 + 3*w - 23],
[997, 997, 4*w^2 - 5*w - 3]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^6 - 10*x^4 + 9*x^2 - 2;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [1, e, -4*e^5 + 38*e^3 - 17*e, -2*e^4 + 19*e^2 - 8, -2*e^5 + 19*e^3 - 9*e, -1, e^3 - 9*e, 9*e^5 - 86*e^3 + 41*e, 3*e^5 - 28*e^3 + 8*e, 3*e^5 - 28*e^3 + 9*e, -e^5 + 9*e^3 - 2*e, 4*e^5 - 39*e^3 + 25*e, 3*e^4 - 28*e^2 + 4, -5*e^5 + 46*e^3 - 9*e, 13*e^4 - 123*e^2 + 56, -6*e^5 + 56*e^3 - 21*e, -3*e^5 + 28*e^3 - 6*e, 10*e^4 - 95*e^2 + 42, 5*e^4 - 46*e^2 + 16, 11*e^4 - 107*e^2 + 52, 3*e^4 - 32*e^2 + 26, 7*e^4 - 66*e^2 + 30, -2*e^4 + 20*e^2 - 16, 11*e^5 - 105*e^3 + 58*e, -9*e^5 + 87*e^3 - 48*e, -2*e^2 + 8, e^4 - 9*e^2 + 12, -16*e^5 + 153*e^3 - 77*e, -13*e^4 + 126*e^2 - 62, -2*e^5 + 23*e^3 - 41*e, -2*e^4 + 17*e^2 + 2, -11*e^4 + 103*e^2 - 50, -2*e^4 + 18*e^2 - 4, 23*e^5 - 221*e^3 + 115*e, -6*e^5 + 55*e^3 - 6*e, -13*e^5 + 123*e^3 - 59*e, -16*e^4 + 150*e^2 - 58, 7*e^5 - 64*e^3 + 13*e, 11*e^5 - 106*e^3 + 55*e, 14*e^4 - 135*e^2 + 68, -10*e^5 + 96*e^3 - 57*e, 33*e^5 - 315*e^3 + 154*e, 9*e^5 - 87*e^3 + 61*e, -7*e^4 + 67*e^2 - 50, 23*e^5 - 220*e^3 + 114*e, -22*e^5 + 211*e^3 - 105*e, e^4 - 7*e^2 - 12, -20*e^5 + 190*e^3 - 82*e, 13*e^4 - 122*e^2 + 46, -e^4 + 6*e^2, 4*e^5 - 35*e^3 - 5*e, 2*e^4 - 15*e^2 - 22, 19*e^5 - 183*e^3 + 103*e, e^5 - 6*e^3 - 36*e, 18*e^4 - 172*e^2 + 68, 6*e^5 - 57*e^3 + 21*e, 19*e^5 - 181*e^3 + 90*e, -24*e^4 + 226*e^2 - 110, 5*e^5 - 51*e^3 + 54*e, 8*e^4 - 72*e^2 + 22, -11*e^4 + 107*e^2 - 48, -14*e^4 + 134*e^2 - 56, -e^4 + 11*e^2 - 4, -29*e^5 + 275*e^3 - 124*e, -5*e^4 + 55*e^2 - 52, -29*e^5 + 278*e^3 - 152*e, -5*e^5 + 49*e^3 - 30*e, -e^3 + 7*e, -5*e^5 + 46*e^3 - e, -26*e^5 + 250*e^3 - 140*e, 16*e^5 - 153*e^3 + 81*e, -5*e^4 + 47*e^2 - 30, 7*e^5 - 63*e^3, -2*e^4 + 19*e^2 - 32, -3*e^3 + 15*e, 3*e^5 - 27*e^3 - 8*e, -4*e^5 + 34*e^3 + 13*e, 5*e^4 - 50*e^2 + 14, -13*e^5 + 126*e^3 - 72*e, 2*e^5 - 22*e^3 + 26*e, -43*e^5 + 410*e^3 - 193*e, 9*e^4 - 85*e^2 + 46, 3*e^5 - 33*e^3 + 64*e, 30*e^5 - 287*e^3 + 143*e, -28*e^4 + 266*e^2 - 126, 23*e^5 - 219*e^3 + 106*e, 28*e^5 - 270*e^3 + 157*e, 21*e^5 - 203*e^3 + 112*e, e^4 - 3*e^2 - 32, -32*e^4 + 308*e^2 - 134, 2*e^4 - 18*e^2 - 6, -10*e^4 + 100*e^2 - 66, 8*e^5 - 79*e^3 + 64*e, -12*e^5 + 119*e^3 - 99*e, 6*e^4 - 54*e^2 + 12, 8*e^4 - 76*e^2 + 30, 14*e^4 - 137*e^2 + 64, 37*e^5 - 354*e^3 + 181*e, -39*e^5 + 372*e^3 - 187*e, -12*e^4 + 114*e^2 - 76, -31*e^4 + 293*e^2 - 142, -16*e^4 + 150*e^2 - 66, -43*e^5 + 409*e^3 - 192*e, 22*e^5 - 212*e^3 + 126*e, 3*e^4 - 31*e^2 + 16, -4*e^4 + 40*e^2 - 20, e^5 - 6*e^3 - 21*e, -2*e^4 + 21*e^2 - 12, -28*e^4 + 270*e^2 - 126, -4*e^4 + 30*e^2 + 14, 3*e^5 - 25*e^3 - 30*e, -7*e^5 + 61*e^3 + 12*e, -e^2 + 20, -33*e^5 + 316*e^3 - 177*e, -35*e^5 + 337*e^3 - 189*e, -e^4 + 16*e^2 - 28, 21*e^4 - 198*e^2 + 96, 31*e^4 - 295*e^2 + 136, -3*e^3 + 27*e, -9*e^4 + 85*e^2 - 36, 38*e^4 - 364*e^2 + 168, 11*e^5 - 113*e^3 + 116*e, 31*e^4 - 293*e^2 + 132, 17*e^4 - 160*e^2 + 62, -31*e^4 + 293*e^2 - 124, -7*e^4 + 61*e^2 - 18, 10*e^5 - 94*e^3 + 26*e, -10*e^4 + 88*e^2 - 6, -5*e^5 + 52*e^3 - 51*e, 6*e^5 - 63*e^3 + 82*e, 5*e^5 - 51*e^3 + 67*e, 41*e^5 - 389*e^3 + 176*e, 16*e^5 - 147*e^3 + 40*e, -39*e^4 + 372*e^2 - 172, 30*e^5 - 282*e^3 + 102*e, -10*e^4 + 96*e^2 - 58, 24*e^4 - 234*e^2 + 118, -17*e^5 + 160*e^3 - 48*e, -28*e^4 + 263*e^2 - 112, -3*e^4 + 27*e^2 - 24, -5*e^5 + 57*e^3 - 93*e, 30*e^4 - 290*e^2 + 136, 32*e^4 - 305*e^2 + 148, 27*e^4 - 257*e^2 + 112, 13*e^4 - 128*e^2 + 88, -24*e^5 + 225*e^3 - 81*e, e^4 - 9*e^2 + 14, -10*e^4 + 98*e^2 - 84, 14*e^5 - 132*e^3 + 38*e, -46*e^5 + 437*e^3 - 206*e, -25*e^5 + 238*e^3 - 110*e, -12*e^4 + 118*e^2 - 58, 11*e^4 - 102*e^2 + 50, 31*e^5 - 291*e^3 + 119*e, -5*e^5 + 47*e^3 - 8*e, 2*e^5 - 18*e^3 - e, 22*e^4 - 210*e^2 + 120, 36*e^5 - 345*e^3 + 183*e, 11*e^5 - 100*e^3 + 25*e, 8*e^5 - 77*e^3 + 41*e, 36*e^5 - 344*e^3 + 189*e, 49*e^5 - 469*e^3 + 254*e, 15*e^4 - 140*e^2 + 46, 30*e^4 - 283*e^2 + 146, -35*e^5 + 339*e^3 - 188*e, -20*e^5 + 195*e^3 - 116*e, 19*e^5 - 187*e^3 + 138*e, 3*e^4 - 33*e^2 + 14, 17*e^5 - 169*e^3 + 146*e];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {17, 17, -2*w^2 + 15}>] := 1;

// EXAMPLE:
// pp := Factorization(2*ZF)[1][1];
// heckeEigenvalues[pp];

print "To reconstruct the Hilbert newform f, type
  f, iso := Explode(make_newform());";

function make_newform();
 M := HilbertCuspForms(F, NN);
 S := NewSubspace(M);
 // SetVerbose("ModFrmHil", 1);
 NFD := NewformDecomposition(S);
 newforms := [* Eigenform(U) : U in NFD *];

 if #newforms eq 0 then;
  print "No Hilbert newforms at this level";
  return 0;
 end if;

 print "Testing ", #newforms, " possible newforms";
 newforms := [* f: f in newforms | IsIsomorphic(BaseField(f), K) *];
 print #newforms, " newforms have the correct Hecke field";

 if #newforms eq 0 then;
  print "No Hilbert newform found with the correct Hecke field";
  return 0;
 end if;

 autos := Automorphisms(K);
 xnewforms := [* *];
 for f in newforms do;
  if K eq RationalField() then;
   Append(~xnewforms, [* f, autos[1] *]);
  else;
   flag, iso := IsIsomorphic(K,BaseField(f));
   for a in autos do;
    Append(~xnewforms, [* f, a*iso *]);
   end for;
  end if;
 end for;
 newforms := xnewforms;

 for P in primes do;
  xnewforms := [* *];
  for f_iso in newforms do;
   f, iso := Explode(f_iso);
   if HeckeEigenvalue(f,P) eq iso(heckeEigenvalues[P]) then;
    Append(~xnewforms, f_iso);
   end if;
  end for;
  newforms := xnewforms;
  if #newforms eq 0 then;
   print "No Hilbert newform found which matches the Hecke eigenvalues";
   return 0;
  else if #newforms eq 1 then;
   print "success: unique match";
   return newforms[1];
  end if;
  end if;
 end for;
 print #newforms, "Hilbert newforms found which match the Hecke eigenvalues";
 return newforms[1];

end function;